Subgroups — C₃×C₅×#1909 #14274943

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#568	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#536	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#419	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#410	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×H(3,ℤ/4)	C₂	1	2	yes	maximal, complemented
6	C₅×#1909	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×#131	—	2	4	no	—
10	C₂×C₃×C₅×M₁₆	—	2	2	no	—
11	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	complemented
13	C₃×C₅×#124	C₄	1	2	yes	complemented
14	C₃×C₅×#123	—	2	2	no	—
15	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
16	C₃×C₅×#117	C₂²	1	2	yes	—
17	C₃×C₅×#117	C₂²	1	1	yes	characteristic
18	C₃×C₅×#103	—	2	4	no	—
19	C₃×C₅×#99	—	2	2	no	—
20	C₃×#1909	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×#568	C₂×C₃	1	1	yes	characteristic
22	C₅×#536	C₂×C₃	1	1	yes	characteristic
23	C₅×#419	C₂×C₃	1	2	yes	complemented
24	C₅×#410	C₂×C₃	1	1	yes	characteristic
25	C₅×H(3,ℤ/4)	C₂×C₃	1	2	yes	complemented
26	C₃×C₅×#41	—	2	2	no	—
27	C₃×C₅×#41	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	—	2	1	no	—
29	C₂×C₃×C₅×Q₈	—	2	2	no	—
30	C₂×C₃×C₅×Q₈	—	2	2	no	—
31	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	—	2	1	no	—
36	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₃×C₅×M₁₆	—	2	4	no	—
39	C₃×C₅×M₁₆	—	2	2	no	—
40	C₃×C₅×M₁₆	—	2	2	no	—
41	C₂×C₃×C₅×C₈	—	2	4	no	—
42	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
43	C₃×C₅×C₄⋊C₄	—	2	4	no	—
44	C₃×C₅×C₄⋊C₄	—	2	1	no	—
45	C₃×C₅×C₄⋊C₄	—	2	2	no	—
46	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
47	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
48	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
49	C₃×C₄²×C₅	—	2	2	no	—
50	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
51	C₃×#568	C₂×C₅	1	1	yes	characteristic
52	C₃×#536	C₂×C₅	1	1	yes	characteristic
53	C₃×#419	C₂×C₅	1	2	yes	complemented
54	C₃×#410	C₂×C₅	1	1	yes	characteristic
55	C₃×H(3,ℤ/4)	C₂×C₅	1	2	yes	complemented
56	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₅×#131	—	2	4	no	—
59	C₂×C₅×M₁₆	—	2	2	no	—
60	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
61	C₅×C₄⋊Q₈	C₃×C₄	1	2	yes	complemented
62	C₅×#124	C₃×C₄	1	2	yes	complemented
63	C₅×#123	—	2	2	no	—
64	C₅×C₂²⋊Q₈	—	2	2	no	—
65	C₅×#117	C₂²×C₃	1	2	yes	—
66	C₅×#117	C₂²×C₃	1	1	yes	characteristic
67	C₅×#103	—	2	4	no	—
68	C₅×#99	—	2	2	no	—
69	#1909	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
70	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
72	C₃×C₅×Q₈	—	2	2	no	—
73	C₃×C₅×Q₈	—	2	4	no	—
74	C₃×C₅×Q₈	—	4	1	no	—
75	C₃×C₅×Q₈	—	4	1	no	—
76	C₃×C₅×D₄	—	2	4	no	—
77	C₃×C₅×D₄	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	4	2	no	—
83	C₂×C₃×C₄×C₅	—	4	2	no	—
84	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	#41	1	2	yes	—
89	C₃×C₅×C₈	—	2	4	no	—
90	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
91	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
92	C₃×#131	—	2	4	no	—
93	C₂×C₃×M₁₆	—	2	2	no	—
94	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
95	C₃×C₄⋊Q₈	C₄×C₅	1	2	yes	complemented
96	C₃×#124	C₄×C₅	1	2	yes	complemented
97	C₃×#123	—	2	2	no	—
98	C₃×C₂²⋊Q₈	—	2	2	no	—
99	C₃×#117	C₂²×C₅	1	2	yes	—
100	C₃×#117	C₂²×C₅	1	1	yes	characteristic
101	C₃×#103	—	2	4	no	—
102	C₃×#99	—	2	2	no	—
103	C₅×#41	—	2	2	no	—
104	C₅×#41	—	2	2	no	—
105	C₂×C₅×Q₈	—	2	1	no	—
106	C₂×C₅×Q₈	—	2	2	no	—
107	C₂×C₅×Q₈	—	2	2	no	—
108	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
109	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
110	C₂×C₅×D₄	C₂×C₃×C₄	1	1	yes	characteristic
111	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
112	C₂²×C₄×C₅	—	2	1	no	—
113	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
114	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
115	C₅×M₁₆	—	2	4	no	—
116	C₅×M₁₆	—	2	2	no	—
117	C₅×M₁₆	—	2	2	no	—
118	C₂×C₅×C₈	—	2	4	no	—
119	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
120	C₅×C₄⋊C₄	—	2	4	no	—
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₄⋊C₄	—	2	2	no	—
123	C₅×C₂²⋊C₄	—	2	4	no	—
124	C₅×C₂²⋊C₄	—	2	1	no	—
125	C₅×C₂²⋊C₄	—	2	2	no	—
126	C₄²×C₅	—	2	2	no	—
127	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
128	#568	C₂×C₃×C₅	1	1	yes	characteristic
129	#536	C₂×C₃×C₅	1	1	yes	characteristic
130	#419	C₂×C₃×C₅	1	2	yes	complemented
131	#410	C₂×C₃×C₅	1	1	yes	characteristic
132	H(3,ℤ/4)	C₂×C₃×C₅	1	2	yes	complemented
133	C₂²×C₃×C₅	—	2	2	no	—
134	C₂²×C₃×C₅	—	4	1	no	—
135	C₂²×C₃×C₅	—	4	2	no	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₃×C₅	#121	1	2	yes	—
138	C₃×C₄×C₅	—	2	2	no	—
139	C₃×C₄×C₅	—	4	2	no	—
140	C₃×C₄×C₅	—	4	2	no	—
141	C₃×C₄×C₅	—	8	2	no	—
142	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
143	C₃×C₄×C₅	#120	1	1	yes	characteristic
144	C₃×C₄×C₅	#123	1	1	yes	characteristic
145	C₃×#41	—	2	2	no	—
146	C₃×#41	—	2	2	no	—
147	C₂×C₃×Q₈	—	2	1	no	—
148	C₂×C₃×Q₈	—	2	2	no	—
149	C₂×C₃×Q₈	—	2	2	no	—
150	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
151	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
152	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
153	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
154	C₂²×C₃×C₄	—	2	1	no	—
155	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
156	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
157	C₃×M₁₆	—	2	4	no	—
158	C₃×M₁₆	—	2	2	no	—
159	C₃×M₁₆	—	2	2	no	—
160	C₂×C₃×C₈	—	2	4	no	—
161	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
162	C₃×C₄⋊C₄	—	2	4	no	—
163	C₃×C₄⋊C₄	—	2	1	no	—
164	C₃×C₄⋊C₄	—	2	2	no	—
165	C₃×C₂²⋊C₄	—	2	4	no	—
166	C₃×C₂²⋊C₄	—	2	1	no	—
167	C₃×C₂²⋊C₄	—	2	2	no	—
168	C₃×C₄²	—	2	2	no	—
169	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
170	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
171	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
172	C₅×Q₈	—	2	2	no	—
173	C₅×Q₈	—	2	4	no	—
174	C₅×Q₈	—	4	1	no	—
175	C₅×Q₈	—	4	1	no	—
176	C₅×D₄	—	2	4	no	—
177	C₅×D₄	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	4	2	no	—
183	C₂×C₄×C₅	—	4	2	no	—
184	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
185	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
187	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₃×#41	1	2	yes	—
189	C₅×C₈	—	2	4	no	—
190	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
191	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
192	#131	—	2	4	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
195	C₄⋊Q₈	C₃×C₄×C₅	1	2	yes	complemented
196	#124	C₃×C₄×C₅	1	2	yes	complemented
197	#123	—	2	2	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	#117	C₂²×C₃×C₅	1	2	yes	—
200	#117	C₂²×C₃×C₅	1	1	yes	characteristic
201	#103	—	2	4	no	—
202	#99	—	2	2	no	—
203	C₂×C₃×C₅	—	2	1	no	—
204	C₂×C₃×C₅	—	2	2	no	—
205	C₂×C₃×C₅	—	4	2	no	—
206	C₂×C₃×C₅	#386	1	1	yes	characteristic
207	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
208	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
209	C₃×Q₈	—	2	2	no	—
210	C₃×Q₈	—	2	4	no	—
211	C₃×Q₈	—	4	1	no	—
212	C₃×Q₈	—	4	1	no	—
213	C₃×D₄	—	2	4	no	—
214	C₃×D₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	4	2	no	—
220	C₂×C₃×C₄	—	4	2	no	—
221	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
222	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
224	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
225	C₂×C₃×C₄	C₅×#41	1	2	yes	—
226	C₃×C₈	—	2	4	no	—
227	C₂²×C₅	—	2	2	no	—
228	C₂²×C₅	—	4	1	no	—
229	C₂²×C₅	—	4	2	no	—
230	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
231	C₂²×C₅	C₃×#121	1	2	yes	—
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	—	4	2	no	—
234	C₄×C₅	—	4	2	no	—
235	C₄×C₅	—	8	2	no	—
236	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
237	C₄×C₅	C₃×#120	1	1	yes	characteristic
238	C₄×C₅	C₃×#123	1	1	yes	characteristic
239	#41	—	2	2	no	—
240	#41	—	2	2	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	C₂×C₃×C₄×C₅	1	2	yes	—
245	C₂×Q₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
246	C₂×D₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
247	C₂×D₄	C₃×C₅×D₄	1	2	yes	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
250	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
251	M₁₆	—	2	4	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	C₃×C₅×D₄	1	4	yes	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
264	C₃×C₅	#1909	1	1	yes	characteristic, direct product factor, Hall
265	C₂²×C₃	—	2	2	no	—
266	C₂²×C₃	—	4	1	no	—
267	C₂²×C₃	—	4	2	no	—
268	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
269	C₂²×C₃	C₅×#121	1	2	yes	—
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	4	2	no	—
272	C₃×C₄	—	4	2	no	—
273	C₃×C₄	—	8	2	no	—
274	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
275	C₃×C₄	C₅×#120	1	1	yes	characteristic
276	C₃×C₄	C₅×#123	1	1	yes	characteristic
277	C₂×C₅	—	2	1	no	—
278	C₂×C₅	—	2	2	no	—
279	C₂×C₅	—	4	2	no	—
280	C₂×C₅	C₃×#386	1	1	yes	characteristic
281	C₂³	C₃×C₅×C₂²⋊C₄	1	2	yes	—
282	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
283	Q₈	—	2	2	no	—
284	Q₈	—	2	4	no	—
285	Q₈	—	4	1	no	—
286	Q₈	—	4	1	no	—
287	D₄	—	2	4	no	—
288	D₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
296	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
297	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
298	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
299	C₂×C₄	C₃×C₅×#41	1	2	yes	—
300	C₈	—	2	4	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	2	2	no	—
303	C₂×C₃	—	4	2	no	—
304	C₂×C₃	C₅×#386	1	1	yes	characteristic
305	C₅	C₃×#1909	1	1	yes	characteristic, direct product factor, Sylow
306	C₂²	—	2	2	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
310	C₂²	C₃×C₅×#121	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	4	2	no	—
313	C₄	—	4	2	no	—
314	C₄	—	8	2	no	—
315	C₄	C₃×C₄×C₅×D₄	1	2	yes	—
316	C₄	C₃×C₅×#120	1	1	yes	characteristic
317	C₄	C₃×C₅×#123	1	1	yes	characteristic
318	C₃	C₅×#1909	1	1	yes	characteristic, direct product factor, Sylow
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	4	2	no	—
322	C₂	C₃×C₅×#386	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	#386
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₃×C₅×#419	C₂	semidirect
5. C₃×C₅×H(3,ℤ/4)	C₂	semidirect
6. C₅×#1909	C₃	direct (N × H)
12. C₃×C₅×C₄⋊Q₈	C₄	semidirect
13. C₃×C₅×#124	C₄	semidirect
20. C₃×#1909	C₅	direct (N × H)
23. C₅×#419	C₂×C₃	semidirect
25. C₅×H(3,ℤ/4)	C₂×C₃	semidirect
53. C₃×#419	C₂×C₅	semidirect
55. C₃×H(3,ℤ/4)	C₂×C₅	semidirect
61. C₅×C₄⋊Q₈	C₃×C₄	semidirect
62. C₅×#124	C₃×C₄	semidirect
69. #1909	C₃×C₅	direct (N × H)
95. C₃×C₄⋊Q₈	C₄×C₅	semidirect
96. C₃×#124	C₄×C₅	semidirect
130. #419	C₂×C₃×C₅	semidirect
132. H(3,ℤ/4)	C₂×C₃×C₅	semidirect
195. C₄⋊Q₈	C₃×C₄×C₅	semidirect
196. #124	C₃×C₄×C₅	semidirect
264. C₃×C₅	#1909	direct (N × H)
305. C₅	C₃×#1909	direct (N × H)
318. C₃	C₅×#1909	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×#568	2
C₃×C₅×#536	2
C₃×C₅×#419	2
C₃×C₅×#410	2
C₃×C₅×H(3,ℤ/4)	2
C₅×#1909	3
C₃×#1909	5

Subgroups — C3×C5×#1909 #14274943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1909 #14274943